There are two kinds of these polynomials. The first kind $T_n$ is defined by the recurrence \begin{align} T_0(x)&=1\\ T_1(x)&=x\\ T_{n+1}(x)&=2xT_n(x)-T_{n-1}(x) \end{align} The second kind $U_n$ is defined by the same recurrence, but with $U_1(x)=2x$.
These polynomials also satisfy the trigonometric identities $$T_n(\cos\theta)=\cos(n\theta)\qquad U_n(\cos\theta)\sin\theta=\sin(n+1)\theta.$$